The set of continuous functions from topological space Y to topological space Z endowed with a topology forms the function space. For A subset of Y, the set of continuous functions from the space A to the space Z forms the underlying function space with an induced topology. The function space has properties of topological space dependent on the properties of the space Z, such as the T₀, T₁, T₂ and T₃ separation axioms. In this paper, we show that the underlying function space inherits the T₀, T₁, T₂ and T₃ separation axioms from the function space, and that these separation axioms are hereditary on function spaces.